Hi,
Pretty much everything is exposed via API. For example, to get a list of recent discussions in ready-to-consume json format, you can just visit (or fetch) this page.
https://discuss.flarum.org/api/discussions
How do you insert it into your homepage depends on your actual platform. It can work even with a static site using client-side javascript only (especially if your forum and the main site are on the same domain).